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54
Efficient MultiView Reconstruction of LargeScale Scenes using Interest Points, Delaunay Triangulation and Graph Cuts
"... We present a novel method to reconstruct the 3D shape of a scene from several calibrated images. Our motivation is that most existing multiview stereovision approaches require some knowledge of the scene extent and often even of its approximate geometry (e.g. visual hull). This makes these approach ..."
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Cited by 56 (6 self)
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We present a novel method to reconstruct the 3D shape of a scene from several calibrated images. Our motivation is that most existing multiview stereovision approaches require some knowledge of the scene extent and often even of its approximate geometry (e.g. visual hull). This makes these approaches mainly suited to compact objects admitting a tight enclosing box, imaged on a simple or a known background. In contrast, our approach focuses on largescale cluttered scenes under uncontrolled imaging conditions. It first generates a quasidense 3D point cloud of the scene by matching keypoints across images in a lenient manner, thus possibly retaining many false matches. Then it builds an adaptive tetrahedral decomposition of space by computing the 3D Delaunay triangulation of the 3D point set. Finally, it reconstructs the scene by labeling Delaunay tetrahedra as empty or occupied, thus generating a triangular mesh of the scene. A globally optimal label assignment, as regards photoconsistency of the output mesh and compatibility with the visibility of keypoints in input images, is efficiently found as a minimum cut solution in a graph.
Manifold reconstruction in arbitrary dimensions using witness complexes
 In Proc. 23rd ACM Sympos. on Comput. Geom
, 2007
"... It is a wellestablished fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, und ..."
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Cited by 39 (11 self)
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It is a wellestablished fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling assumptions. Unfortunately, these results do not extend to higherdimensional manifolds, even under stronger sampling conditions. In this paper, we show how the sets of witnesses and landmarks can be enriched, so that the nice relations that exist between both complexes still hold on higherdimensional manifolds. We also use our structural results to devise an algorithm that reconstructs manifolds of any arbitrary dimension or codimension at different scales. The algorithm combines a farthestpoint refinement scheme with a vertex pumping strategy. It is very simple conceptually, and it does not require the input point sample W to be sparse. Its time complexity is bounded by c(d)W  2, where c(d) is a constant depending solely on the dimension d of the ambient space. 1
Dense Point Sets Have Sparse Delaunay Triangulations
"... Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms ..."
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Cited by 29 (2 self)
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Delaunay triangulations and Voronoi diagrams are one of the most thoroughly studies objects in computational geometry, with numerous applications including nearestneighbor searching, clustering, finiteelement mesh generation, deformable surface modeling, and surface reconstruction. Many algorithms in these application domains begin by constructing the Delaunay triangulation or Voronoi diagram of a set of points in R³. Since threedimensional Delaunay triangulations can have complexity Ω(n²) in the worst case, these algorithms have worstcase running time \Omega (n2). However, this behavior is almost never observed in practice except for highlycontrived inputs. For all practical purposes, threedimensional Delaunay triangulations appear to have linear complexity. This frustrating
An Empirical Comparison of Techniques for Updating Delaunay Triangulations
, 2004
"... The computation of Delaunay triangulations from static point sets has been extensively studied in computational geometry. When the points move with known trajectories, kinetic data structures can be used to maintain the triangulation. However, there has been little work so far on how to maintain the ..."
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Cited by 29 (2 self)
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The computation of Delaunay triangulations from static point sets has been extensively studied in computational geometry. When the points move with known trajectories, kinetic data structures can be used to maintain the triangulation. However, there has been little work so far on how to maintain the triangulation when the points move without explicit motion plans, as in the case of a physical simulation. In this paper we examine how to update Delaunay triangulations after small displacements of the defining points, as might be provided by a physicsbased integrator. We have implemented a variety of update algorithms, many new, toward this purpose. We ran these algorithms on a corpus of data sets to provide running time comparisons and determined that updating Delaunay can be significantly faster than recomputing.
Delaunay Deformable Models: Topologyadaptive Meshes Based on the Restricted Delaunay triangulation
, 2006
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SpatioTemporal Shape from Silhouette using FourDimensional Delaunay Meshing
, 2007
"... We propose a novel method for computing a fourdimensional (4D) representation of the spatiotemporal visual hull of a dynamic scene, based on an extension of a recent provably correct Delaunay meshing algorithm. By considering time as an additional dimension, our approach exploits seamlessly the tim ..."
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Cited by 26 (3 self)
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We propose a novel method for computing a fourdimensional (4D) representation of the spatiotemporal visual hull of a dynamic scene, based on an extension of a recent provably correct Delaunay meshing algorithm. By considering time as an additional dimension, our approach exploits seamlessly the time coherence between different frames to produce a compact and highquality 4D mesh representation of the visual hull. The 3D visual hull at a given time instant is easily obtained by intersecting this 4D mesh with a temporal plane, thus enabling interpolation of objects’ shape between consecutive frames. In addition, our approach offers easy and extensive control over the size and quality of the output mesh as well as over its associated reprojection error. Our numerical experiments demonstrate the effectiveness and flexibility of our approach for generating compact, highquality, timecoherent visual hull representations from real silhouette image data.
VietorisRips Complexes also Provide Topologically Correct Reconstructions of Sampled Shapes
, 2012
"... Given a point set that samples a shape, we formulate conditions under which the Rips complex of the point set at some scale reflects the homotopy type of the shape. For this, we associate with each compact set X of Rn two realvalued functions cX and hX defined on R+ which provide two measures of ho ..."
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Cited by 22 (8 self)
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Given a point set that samples a shape, we formulate conditions under which the Rips complex of the point set at some scale reflects the homotopy type of the shape. For this, we associate with each compact set X of Rn two realvalued functions cX and hX defined on R+ which provide two measures of how much the set X fails to be convex at a given scale. First, we show that, when P is a finite point set, an upper bound on cP (t) entails that the Rips complex of P at scale r collapses to the Čech complex of P at scale r for some suitable values of the parameters t and r. Second, we prove that, when P samples a compact set X, an upper bound on hX over some interval guarantees a topologically correct reconstruction of the shape X either with a Čech complex of P or with a Rips complex of P. Regarding the reconstruction with Čech complexes, our work compares well with previous approaches when X is a smooth set and surprisingly enough, even improves constants when X has a positive µreach. Most importantly, our work shows that Rips complexes can also be used to provide shape reconstructions having the correct homotopy type. This may be of some computational interest in high dimensions.
Complexity of Delaunay triangulation for points on lowerdimensional polyhedra
 PROC. 18TH ANNU. ACMSIAM SYMPOS. DISCRETE ALGO
, 2007
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The implicit structure of ridges of a smooth parametric surface
 INRIA
, 2005
"... Given a smooth surface, a blue (red) ridge is a curve such that at each of its points, the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching ..."
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Cited by 13 (3 self)
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Given a smooth surface, a blue (red) ridge is a curve such that at each of its points, the maximum (minimum) principal curvature has an extremum along its curvature line. Ridges are curves of extremal curvature and therefore encode important informations used in segmentation, registration, matching and surface analysis. State of the art methods for ridge extraction either report red and blue ridges simultaneously or separately —in which case a local orientation procedure of principal directions is needed, but no method developed so far certifies the topology of the curves reported. On the way to developing certified algorithms independent from local orientation procedures, we make the following fundamental contributions. For any smooth parametric surface, we exhibit the implicit equation P = 0 of the singular curve P encoding all ridges and umbilics of the surface (blue and red), and show how to recover the colors from factors of P. Exploiting second order derivatives of the principal curvatures, we also derive a zero dimensional system coding the socalled turning points, from which elliptic and hyperbolic ridge sections of the two colors can be derived. Both contributions exploit properties of the Weingarten map of the surface in the specific parametric setting and require computer algebra. Algorithms exploiting the structure of P for algebraic surfaces are developed in a companion paper.